I would tend to agree with most on the August dates being preferable for your purposes, but be aware of the summer "monsoon season" up here as well. During August, we have frequent afternoon thunderstorms that only last a few minutes, but you definitely do not want to be on an exposed mountain face in a thunderstorm.....
For Rocky Mountain National Park--consider staying on the west side in Grand Lake instead of the more commercial Estes Park at the eastern entrance. There are fewer amenities in Grand Lake, but similar activities--and FAR fewer people. Plus, there is a great hike up to the headwaters of the Colorado River on that side of the park. Estes Park is still an easy day trip over Trail Ridge Road right through the park, so you can still spend a day over there as well.
This past summer I stayed at the Elk Creek Campground in Grand Lake, and it was basic, but very nice. There were also moose wandering through the campground, so you do have to be careful about that.
Last fall I made my first trip ever up to Yellowstone and Grand Teton. I stayed at the Grand Teton Park RV Resort at Moran Junction, and the park itself was nice, but I was not thrilled with the location. Moran Junction really doesn't have anything OTHER than an entrance station for the national parks and a couple of campgrounds, so although it was a central location, it really wasn't CONVENIENT to anything. The nearest town was Jackson, WY, which is about 30 miles away--not the most convenient for a quick grocery store run. I think the next time I go up there, I will probably spend part of the time in the Jackson area to visit GTNP, and then maybe move to a campground in some place like West Yellowstone, MT, to visit Yellowstone. I just ran into an issue with the parks being SOOOO spread out that it made for a lot of driving. The day I went up to Mammoth Hot Springs and Gardiner, MT, I realized at about 4pm that I still had about a 3-4 hour drive (NOT including stops) to get back to my camper at Moran Junction. It put a bit of a damper on the idea of stopping to take in the sights on the way back to the campground......